Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
What makes nonholonomic integrators work?
Klas Modin
Collaborator: Olivier Verdier
Constrained Lagrangian systems
Lagrange-d'Alembert
\displaystyle\delta \int_a^b L(\mathbf{q}(t),\dot{\mathbf{q}}(t))dt = 0
for virtual displacements \( \delta\mathbf{q}\)
A(\mathbf{q}(t))\delta\mathbf{q}(t) = 0
Constraint defined by distribution \(A(\mathbf{q})\dot{\mathbf{q}}=0\)
Constrained Lagrangian systems
Lagrange-d'Alembert
\displaystyle\delta \int_a^b L(\mathbf{q}(t),\dot{\mathbf{q}}(t))dt = 0
for virtual displacements \( \delta\mathbf{q}\)
A(\mathbf{q}(t))\delta\mathbf{q}(t) = 0
Constraint defined by distribution \(A(\mathbf{q})\dot{\mathbf{q}}=0\)
Holonomic systems
\displaystyle A(\mathbf q)\dot{\mathbf q} = \frac{d}{dt}\mathbf F(\mathbf q)
\mathcal C = \{ \mathbf{q}\in \mathcal Q \mid \mathbf F(\mathbf q) = 0 \}
Distribution is integrable
\(\Rightarrow\) Lagrangian dynamics on \(T\mathcal C\)
Nonholonomic systems
\displaystyle A(\mathbf q)\dot{\mathbf q} \neq \frac{d}{dt}\mathbf F(\mathbf q)
\mathcal M = \text{state space}
Distribution nonintegrable
\mathbf q
\dot{\mathbf q}
\mathbf q\cdot\dot{\mathbf q} = 0
Holonomic systems
\displaystyle A(\mathbf q)\dot{\mathbf q} = \frac{d}{dt}\mathbf F(\mathbf q)
\mathcal C = \{ \mathbf{q}\in \mathcal Q \mid \mathbf F(\mathbf q) = 0 \}
Distribution is integrable
\(\Rightarrow\) Lagrangian dynamics on \(T\mathcal C\)
Nonholonomic systems
\displaystyle A(\mathbf q)\dot{\mathbf q} \neq \frac{d}{dt}\mathbf F(\mathbf q)
\mathcal M = \text{state space}
Distribution nonintegrable
\mathbf q
\dot{\mathbf q}
\mathbf q\cdot\dot{\mathbf q} = 0
Nonholonomic examples
Rolling disk
Nonholonomic examples
Knife edge
Nonholonomic examples
Continuously variable transmission (CVT)
What makes holonomic integrators work?
Definition: Numerical integrator
Map \( \Phi_h: (\mathbf{q}_{k},\dot{\mathbf{q}}_{k})\mapsto (\mathbf{q}_{k+1},\dot{\mathbf{q}}_{k+1}) \) that preserves constraints and approximates exact flow \(\varphi^h\)
Exact flow \(\varphi^h\) \(\Rightarrow\) Hamiltonian system on \(T^*\mathcal C\)
Geometric numerical integration and backward error analysis:
- Integrator is exact flow of modified system on \(T^*\mathcal C\)
- Modified system Hamiltonian?
\(\Rightarrow\) modified Hamiltonian preserved - Original system Arnold-Liouville integrable?
\(\Rightarrow\) KAM stability of tori
What about nonholonomic integrators?
Exact flow \(\varphi^h\) \(\Rightarrow\) energy system on \(\mathcal M\)
- Is that all?
- Not clear what "structure preserving" means
Idea: discrete analog of Lagrange d'Alembert (DLA)
Energy
Time
Why so good?
Notion: additional structure behind long-time behaviour
Strategy
- Identify structure of standard nonholonomic test problems
Clues
- Reversibility known to be important
[McLachlan and Perlmutter, 2006]
- Most test problems integrable
(First integrals common evaluation criterion )
Notion: additional structure behind long-time behaviour
Strategy
- Identify structure of standard nonholonomic test problems
Clues
- Reversibility known to be important
[McLachlan and Perlmutter, 2006]
- Most test problems integrable
(First integrals common evaluation criterion )
Energy
Time
DLA but nonreversible
DLA and reversible
Nonholonomically coupled systems
Definition: Nonholonomically Coupled System (NCS)
L(\mathbf x, \xi, \dot{\mathbf x},\dot\xi) = L_1(\mathbf x,\dot{\mathbf x}) + L_2(\xi,\dot\xi)
A(\xi)\dot{\mathbf x} = 0
\displaystyle\frac{d}{dt}\frac{\partial L_2}{\partial\dot\xi} =\frac{\partial L_2}{\partial \xi}
independent subsystem (driver)
\dot{\mathbf x} = \mathbf{k}(\xi) v \qquad \dot v= \mathbf{k}(\xi)\cdot \mathbf F(\mathbf x)
Conserved energies: \(E_1(\mathbf x,v)\) and \(E_2(\xi,\dot\xi)\)
Is system on \(\mathcal M\) integrable?
Integrable NCS
\dot{\mathbf u} = B(\xi)\mathbf u
\(B(\xi)\in \mathfrak{g}\subset \mathfrak{gl}(n)\)
(\(\mathfrak{g}\)-system)
Definition: ODE system integrable \(\iff\) Action-Angle variables
Theorem (M. and Verdier)
NCS (with additional assumptions) are fibrated over integrable system
Reduced dynamics
Next step: KAM theory
Requirement for KAM:
- dynamics either symplectic or reversible
Exact flow:
(\mathbf q_0,\dot{\mathbf q}_0)\mapsto (\mathbf q(h),\dot{\mathbf q}(h))
\mathcal M \to \mathcal M
\mathcal N \to \mathcal N
Action-Angle variables
\mathcal N \to \mathcal N
KAM stable tori
reversible perturbation
Discrete flow:
(\mathbf q_k,\dot{\mathbf q}_k)\mapsto (\mathbf q_{k+1},\dot{\mathbf q}_{k+1})
\mathcal M \to \mathcal M
\approx
Next step: KAM theory
Requirement for KAM:
- dynamics either symplectic or reversible
Exact flow:
(\mathbf q_0,\dot{\mathbf q}_0)\mapsto (\mathbf q(h),\dot{\mathbf q}(h))
\mathcal M \to \mathcal M
\mathcal N \to \mathcal N
Action-Angle variables
\mathcal N \to \mathcal N
KAM stable tori
reversible perturbation
Discrete flow:
(\mathbf q_k,\dot{\mathbf q}_k)\mapsto (\mathbf q_{k+1},\dot{\mathbf q}_{k+1})
\mathcal M \to \mathcal M
\approx
possible failure
Next step: KAM theory
Requirement for KAM:
- dynamics either symplectic or reversible
Exact flow:
(\mathbf q_0,\dot{\mathbf q}_0)\mapsto (\mathbf q(h),\dot{\mathbf q}(h))
\mathcal M \to \mathcal M
\mathcal N \to \mathcal N
Action-Angle variables
\mathcal N \to \mathcal N
KAM stable tori
reversible perturbation
Discrete flow:
(\mathbf q_k,\dot{\mathbf q}_k)\mapsto (\mathbf q_{k+1},\dot{\mathbf q}_{k+1})
\mathcal M \to \mathcal M
\approx
possible failure
Theorem (M. and Verdier)
If discrete flow
- Preserve fibration
- Preserves reversibility (from action-angle variables)
then all integrals are nearly conserved
Next step: KAM theory
Requirement for KAM:
- dynamics either symplectic or reversible
Exact flow:
(\mathbf q_0,\dot{\mathbf q}_0)\mapsto (\mathbf q(h),\dot{\mathbf q}(h))
\mathcal M \to \mathcal M
\mathcal N \to \mathcal N
Action-Angle variables
\mathcal N \to \mathcal N
KAM stable tori
reversible perturbation
Discrete flow:
(\mathbf q_k,\dot{\mathbf q}_k)\mapsto (\mathbf q_{k+1},\dot{\mathbf q}_{k+1})
\mathcal M \to \mathcal M
\approx
possible failure
Theorem (M. and Verdier)
If discrete flow
- Preserve fibration
- Preserves reversibility (from action-angle variables)
then all integrals are nearly conserved
Numerical verification
CVT problem
Sub-energy \(E_1\)
Time
Is the result sharp?
Strategy: construct NCS by perturbing
- fibration projection
- reversibility map
IMPORTANT: resulting systems are still integrable NCS
Knife edge, perturbed fibration
Total energy
Is the result sharp?
Strategy: construct NCS by perturbing
- fibration projection
- reversibility map
IMPORTANT: resulting systems are still integrable NCS
CVT, perturbed reversibility
Sub-energy \(E_1\)
Conclusion and outlook
- DLA does not imply structure preservation
-
Reason: class of nonholonomic systems too large for generic DLA approach
-
Strategy: focus on subclasses with enough structure for e.g. KAM (Hamiltonian or reversible)
Already started: recent work by [Martín de Diego et. al.]
THANKS!
Slides available at: slides.com/kmodin
What makes nonholonomic integrators work?
By Klas Modin
What makes nonholonomic integrators work?
Online-presentation given 2021-05 in the Geometry, Dynamics, and Mechanics Seminar.
